Multiplying and Dividing Fractions and Mixed Numbers
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1 Multiplying and Dividing Fractions and Mixed Numbers When two fractions are multiplied, the result is another fraction with a denominator that is equal to the product of the denominators of the two fractions. Also, the numerator of the product consists of the product of the numerators of the two fractions. This can be illustrated with the following example. Example What is? The expression means of because the word of implies multiplication when used with fractions or percents. of may be represented with pizza slices. When we try to select of of a pizza the result is an uneven number of eighths. In order to obtain whole pieces, it is necessary to slice the pizza into equal pieces and then the result is whole pieces. The resulting fraction is. of = of = MATH FACT The word of usually means multiply when used in fraction and percent problems. Example: of = Example: 0% of = 0% = = In Example, the product of the two fractions could have been obtained by multiplying the numerators and multiplying the denominators as shown here.
2 0 0 0 ; however, reduces down to. 0 0 In a similar manner, the product is obtained by multiplying the numerators, and by multiplying the denominators. When multiplying fractions, factors common to any numerator and any denominator may be canceled out before multiplying. Note that a factor is any number that evenly divides into the numerator or the denominator. may be multiplied out after canceling, as is 0 shown here. = 0 = In this step, the factor of is divided out of and out of. 0 0 = = = A factor of is again divided out of and PROCEDURE TO MULTIPLY FRACTIONS. Cancel out factors by dividing out factors common to any numerator and any denominator.. Multiply the numerators together.. Multiply the denominators together.
3 0 0 Example Multiply. 0 0 = = = = 0 0 In this example, a factor of was divided out of and. Then, a factor of 0 was divided out of 0 and 0. The remaining numerators and denominators were multiplied out to result in. 0 Example Multiply. = 0 0 = = 0 0 = = In this example, a factor of was divided out of and ; a factor of was divided out of and ; a factor of was divided out of and 0; and the factors of were divided out and canceled. Example Multiply. Since no common factors cancel, the product is =. 0
4 A B C C A Example Multiply out the fractions where A, B, and C represent real numbers. A A B C C C A First note that the fractions may be written as. A factor of A may be divided out of the numerator and the denominator. Also, a factor of C may be divided out of the numerator and the denominator. The result is A A B C C C A A A B C C A B C = = = A B C C A This result may also be written as ABC. Multiplying Mixed Numbers The procedure for multiplying mixed numbers is straightforward. First, convert each mixed number into an improper fraction. Then, multiply the improper fractions using the procedure for fraction multiplication. This procedure is restated below. PROCEDURE FOR MULTIPLYING MIXED NUMBERS. Convert all mixed numbers into improper fractions.. Multiply the fractions using the procedure for fraction multiplication. Example Multiply. The mixed numbers are converted into improper fractions. = = =
5 .Example Multiply. 0 The mixed numbers are converted into improper fractions. 0 0 = = = Applications of Fraction Multiplication In everyday situations, we use fraction multiplication in a variety of applications. For example, if we want to calculate what half of gallons is, we multiply to obtain a result of. More applications of fraction multiplication are given below. Example What is of 0? First, realize that the word of means multiply when used in fraction and percent applications. Thus, 0 0 of 0 = 0 = = or. In general, the word of can be replaced with a multiply sign in most fraction and percent problems. MATH REMINDER The word of usually means multiply when used in fraction and percent problems. Example: of 000 = 000 Example: 0% of 00 = 0% 00
6 Example A doctor reduces the amount of medication a patient takes. The regular dosage was 00 milligrams but the patient now needs to take of the regular dosage. How many milligrams of the medication should the patient take? of the regular dosage = of 00 = = = = 0 milligrams Example In a given district, of the voters voted for a particular candidate. If there are one-half of a million voters in the district, how many of them voted for the candidate?,000,000 of the voters = of of million =,000,000,000,000 = =,000 voters Division of Fractions When dividing by a fraction, the net result is multiplication by the reciprocal of that fraction. The reciprocal of any given fraction is obtained by switching the numerator and the denominator with each other. When we write the reciprocal of a fraction we are said to be inverting the fraction. For example, or. In this example, the division of by means How many eighths are in? Since is equal to, the answer is. In another example, = = or. In this second example, the division of by means Divide in half. Half of is =.
7 PROCEDURE FOR DIVIDING FRACTIONS. Invert the fraction you are dividing by. The inverted fraction is called the reciprocal.. Multiply the first fraction by the second inverted fraction. 0 Example Divide. 0 0 = = 0 Note that if a factor of were canceled before inverting, an incorrect result of would have been obtained. Example Divide. = Note that may be written as. = = 0 Example Write as a single fraction. This fraction consists of fractions in the numerator and the denominator. The fraction bar in the center indicates division. Thus, this expression is equal to. 0 = = = which reduces to.
8 A B A C A A A C A C C C = = = = B C B A B A B B Example Divide where A, B, and C represent real numbers. In this example, the second fraction was inverted and multiplied by the first. The factor A was divided out of the numerator and the denominator and the resulting numerators and denominators were multiplied out. Dividing Mixed Numbers The procedure for dividing mixed numbers requires that the mixed numbers be converted into improper fractions. The procedure is restated here. PROCEDURE FOR DIVIDING MIXED NUMBERS. Convert all mixed numbers into improper fractions.. Divide the improper fractions using the procedure for fraction division. (Invert and multiply) Example Divide. = = = Example Divide is written as. = = =
9 Example Rewrite the fraction by dividing by. = = = =. Example Rewrite the fraction by dividing by. = = = =. Example Rewrite the fraction by dividing by. = = = =.
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